# nLab complex volume

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

#### Differential cohomology

differential cohomology

# Contents

## Idea

The Cheeger-Simons classes are complexified secondary invariants.

Under identifying the fundamental class of a hyperbolic 3-manifold $X$ as an element in the Bloch group, the corresponding degree-3 Cheegers-Simons invariant is the complex volume of the 3-manifold, namely the linear combination

$CS + i vol$

of its Chern-Simons invariant and its volume (e.g. Neumann 11, section 2.3, Garoufalidis-Thurston-Zickert 11).

This appears as the action in analytically continued Chern-Simons theory.

(It is, incidentally, also the contribution of a corresponding membrane instanton wrapping a hyperbolic 3-cycle.)

## Properties

### Volume conjecture

The volume conjecture for the Reshetikhin-Turaev construction states that in the classical limit it converges to the complex volume (MMOTY 02, Conjedtcure 1.2, see also Chen-Yang 15)

The volume conjecture (Kashaev’s conjecture) for complex volume is due to

• Varghese Mathai, section 6 of $L^2$-analytic torsion, Journal of Functional Analysis Volume 107, Issue 2, 1 August 1992, Pages 369–386